I present to you the free version of MailStore. It will fetch mail from your accounts using POP3 or IMAP4 and store them on your hard disk for searching and reading. It comes ready to access many popular servers such as Hotmail and Gmail, and can export to text files or certain email client database formats.
Of course this won't let you read new mail while the servers are down, but at least your old messages will be there.
Free, easy to use, small learning curve. The rest will become a habit that you won't regret the next time you'll need to find that bit of info that you didn't write down when your email goes offline.
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